FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE

FUZZY ARITHMETIC
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Published Date:25-10-2017
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CHAPTER 12 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE Said the Mock Turtle with a sigh, ‘‘I only took the regular course.’’ ‘‘What was that?’’ inquired Alice. ‘‘Reeling and Writhing, of course, to begin with,’’ the Mock Turtle replied; ‘‘and the different branches of Arithmetic – Ambition, Distraction, Uglification, and Derision.’’ LewisCarroll Alice in Wonderland,1865 As Lewis Carroll so cleverly implied as early as 1865 (he was, by the way, a brilliant mathematician), there possibly could be other elements of arithmetic: consider those of ambition,distraction,uglification,andderision.Certainlyfuzzylogichasbeendescribedin worse terms by many people over the last four decades Perhaps Mr. Carroll had a presage of fuzzy set theory exactly 100 years before Dr. Zadeh; perhaps, possibly. In this chapter we see that standard arithmetic and algebraic operations, which are basedafterallonthefoundationsofclassicalsettheory,canbeextendedtofuzzyarithmetic and fuzzy algebraic operations. This extension is accomplished with Zadeh’s extension principle Zadeh, 1975. Fuzzy numbers, briefly described in Chapter 4, are used here because such numbers are the basis for fuzzy arithmetic. In this context the arithmetic operationsarenotfuzzy;thenumbersonwhichtheoperationsareperformedarefuzzyand, hence, so too arethe results of these operations. Conventional interval analysis is reviewed as a prelude to some improvements and approximations to the extension principle, most notably the fuzzy vertex method and its alternative forms. EXTENSIONPRINCIPLE In engineering, mathematics, and the sciences, functions are ubiquitous elements in modeling. Consider a simple relationship between one independent variable and one Fuzzy Logic with Engineering Applications, Second Edition T. J. Ross  2004 John Wiley & Sons, Ltd ISBNs: 0-470-86074-X (HB); 0-470-86075-8 (PB) www.MatlabSite.com446 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE FIGURE12.1 x f(x) y A simple single-input,single-outputmapping (function). dependent variable as shown in Fig. 12.1. This relationship is a single-input, single-output process where the transfer function (the box in Fig. 12.1) represents the mapping provided by the general function f. In the typical case, f is of analytic form, e.g., y = f(x),the input, x, is deterministic, and the resulting output, y, is also deterministic. How can we extend this mapping to the case where the input, x, is a fuzzy variable or a fuzzyset, and wherethe function itself could be fuzzy?That is, how canwe determine the fuzziness in the output, y, based on either a fuzzy input or a fuzzy function or both (mapping)? An extension principle developed by Zadeh 1975 and later elaborated by Yager 1986 enables us to extend the domain of a function on fuzzy sets. The material of the next several sections introduces the extension principle by first reviewing theoretical issues of classical (crisp) transforms, mappings, and relations. The theoretical material then moves to the case where the input is fuzzy but the function itself is crisp, then to the case where the input and the function both are fuzzy. Simple examples to illustrate the ideas are provided. The next section serves as a more practical guide to the implementationoftheextensionprinciple,withseveralnumericalexamples.Theextension principle is a very powerful idea that, in many situations, provides the capabilities of a ‘‘fuzzy calculator.’’ CrispFunctions,Mapping,andRelations Functions (also called transforms), such as the logarithmic function, y = log(x),orthe linear function y = ax + b, are mappings from one universe, X, to another universe, Y. Symbolically, this mapping (function, f) is sometimes denoted f :X → Y.Other terminology calls the mapping y = f(x) the image of x under f, and the inverse mapping, −1 x = f (y),istermedthe original image of y. A mapping can also be expressed by a relationR(asdescribedinChapter 3),ontheCartesianspaceX ×Y.Sucharelation(crisp) can be described symbolically as R=(x, y) y = f(x), with the characteristic function describing membership of specific x,y pairs to the relation R as  1,y = f(x) χ (x, y) = (12.1) R 0,y = f(x) Now, since we can define transform functions, or mappings, for specific elements of one universe (x) to specific elements of another universe (y),wecanalsodothe same thing for collections of elements in X mapped to collections of elements in Y. Such collections have been referred to in this text as sets. Presumably, then, all possible sets in the power set of X can be mapped in some fashion (there may be null mapping for many of the combinations) to the sets in the power set of Y, i.e., f :P(X) → P(Y).Foraset A defined on universe X, its image, set B on the universe Y, is found from the mapping, B = f(A)=y for all x ∈ A,y = f(x), where B will be defined by its characteristic value  χ (y) = χ (y) = χ (x) (12.2) B f(A) A y=f(x) www.MatlabSite.comEXTENSION PRINCIPLE 447 Example12.1. Suppose we have a crisp set A=0,1, or, using Zadeh’s notation,   0 0 1 1 0 A = + + + + −2 −1 0 1 2 defined on the universe X=−2,−1,0,1,2 and a simple mapping y =4x+2. We wish to find the resulting crisp set B on an output universe Y using the extension principle. From the mapping we can see that the universe Y will be Y=2,6,10. The mapping described by Eq. (12.2) will yield the following calculations for the membership values of each of the elements in universe Y: χ (2)=∨χ (0)= 1 B A χ (6)=∨χ (−1), χ (1)=∨0,1= 1 B A A χ (10)=∨χ (−2), χ (2)=∨0,0= 0 B A A Notice there is only one way to get the element 2 in the universe Y, but there are two ways to get the elements 6 and 10 in Y. Written in Zadeh’s notation this mapping results in the output   1 1 0 B = + + 2 6 10 or, alternatively,B=2,6. Suppose we want to find the image B on universe Y using a relation that expresses the mapping. This transform can be accomplished by using the composition operation described in Chapter 3 for finite universe relations, where the mapping y = f(x) is a general relation. Again, for X=−2,−1,0,1,2 and a generalized universe Y=0,1,...,9,10,thecrisp relation describing this mapping (y =4x+2) is 0 12 345 678 910   −2 0 00 000 000 0 1 −10 00 000 100 0 0    R = 0 0 01 000 000 0 0     1 0 00 000 100 0 0 2 0 00 000 000 0 1 ◦ The image B can be found through composition (since X and Y are finite): that is, B = A R (we noteherethatanyset,sayA,can beregarded asa one-dimensionalrelation),where, again using Zadeh’s notation,   0 0 1 1 0 A = + + + + −2 −1 0 1 2 and B is found by means of Eq. (3.9) to be  1, for y = 2,6 χ (y) = (χ (x) ∧ χ (x, y)) = B A R 0, otherwise x∈X or in Zadeh’s notation on Y,   0 0 1 0 0 0 1 0 0 0 0 B = + + + + + + + + + + 0 1 2 3 4 5 6 7 8 9 10 FunctionsofFuzzySets–ExtensionPrinciple Again we start with two universes of discourse, X and Y, and a functional transform (mapping) of the form y = f(x). Now suppose that we have a collection of elements in www.MatlabSite.com448 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE universe x thatformafuzzysetA.WhatistheimageoffuzzysetAonXunderthemapping ∼ ∼ f?Thisimagewillalsobefuzzy,saywedenoteitfuzzysetB;anditwillbefoundthrough ∼ the same mapping, i.e., B = f(A). ∼ ∼ The membership functions describing A and B will now be defined on the universe ∼ ∼ of a unit interval 0,1, and for the fuzzy case Eq. (12.2) becomes  µ (y) = µ (x) (12.3) B A ∼ ∼ f(x)=y A convenient shorthand for many fuzzy calculations that utilize matrix relations involvesthe fuzzy vector.Basically,afuzzyvectorisavectorcontainingfuzzymembership values.SupposethefuzzysetAisdefinedon nelementsinX,forinstanceon x ,x ,...,x , 1 2 n ∼ and fuzzy set B is defined on m elements in Y, say on y ,y ,...,y . The array of 1 2 m ∼ membership functions for each of the fuzzy sets A and B can then be reduced to fuzzy ∼ ∼ vectors by the following substitutions: a=a ,...,a =µ (x ),...,µ (x )=µ (x ),for i = 1,2,...,n (12.4) 1 n A 1 A n A i ∼ ∼ ∼ ∼ b=b ,...,b =µ (y ),...,µ (y )=µ (y ),for j = 1,2,...,m (12.5) 1 m B 1 B m B j ∼ ∼ ∼ ∼ Now, the image of fuzzy set A can be determined through the use of the composition ∼ ◦ ◦ operation,orB = A R,orwhenusingthefuzzyvectorform,b = a RwhereRisan n × m ∼ ∼ ∼ ∼ ∼ ∼ ∼ fuzzy relation matrix. More generally,suppose our input universe comprises the Cartesianproduct ofmany universes. Then the mapping f is defined on the power sets of this Cartesian input space and the output space, or f :P(X ×X ×···×X ) −→ P(Y)(12.6) 1 2 n Let fuzzy sets A ,A ,...,A be defined on the universes X ,X ,...,X . The mapping 1 2 n 1 2 n ∼ ∼ ∼ for these particular input sets can now be defined as B = f(A ,A ,...,A ),wherethe 1 2 n ∼ ∼ ∼ ∼ membership function of the image B is given by ∼ µ (y) = max minµ (x ), µ (x ),...,µ (x ) (12.7) B A 1 A 2 A n 1 2 n ∼ ∼ ∼ ∼ y=f(x ,x ,...,x ) 1 2 n In the literature Eq. (12.7) is generally called Zadeh’s extension principle. Equation (12.7) is expressed for a discrete-valued function, f. If the function, f, is a continuous-valued expression, the max operator is replacedby the sup (supremum) operator(the supremum is the least upper bound). FuzzyTransform(Mapping) The material presented in the preceding two sections is associated with the issue of ‘‘extending’’ fuzziness in an input set to an output set. In this case, the input is fuzzy, the output is fuzzy, but the transform (mapping) is crisp, or f : A → B. What happens in a ∼ ∼ more restricted case where the input is a single element (a nonfuzzy singleton) and this single element maps to a fuzzy set in the output universe? In this case the transform, or mapping, is termed a fuzzy transform. www.MatlabSite.comEXTENSION PRINCIPLE 449 Formally, let a mapping exist from an element x in universe X (x ∈ X) to a fuzzy set BinthepowersetofuniverseY,P(Y).Suchamappingiscalledafuzzymapping,f,where ∼ ∼ the output is no longer a single element, y, but a fuzzy set B, i.e., ∼ B = f(x) (12.8) ∼ ∼ IfX andY arefinite universes,thefuzzymapping expressedin Eq. (12.8)canbedescribed as a fuzzy relation, R, or, in matrix form, ∼ y y ... y ... y 1 2 j m   x r r ... r ... r 1 11 12 1j 1m   x r r ... r ... r 2 21 22 2j 2m   R = (12.9)   ∼ x r r ... r ... r i i1 i2 ij im x r r ... r ... r n n1 n2 nj nm Foraparticularsingleelementoftheinputuniverse,say x ,itsfuzzyimage,B = f(x ), i i i ∼ ∼ is given in a general symbolic form as µ (y ) = r (12.10) B j ij i ∼ or, in fuzzy vector notation, b =r ,r ,...,r (12.11) i1 i2 im ∼i Hence, the fuzzy image of the element x is given by the elements in the ith row of the i fuzzy relation, R, defining the fuzzy mapping, Eq. (12.9). ∼ Supposewenowfurthergeneralizethesituationwhereafuzzyinputset,sayA,maps ∼ to a fuzzy output through a fuzzy mapping, or B = f(A)(12.12) ∼ ∼ ∼ The extension principle again can be used to find this fuzzy image, B, by the following ∼ expression:  µ (y) = (µ (x) ∧ µ (x, y)) (12.13) B A R ∼ ∼ ∼ x∈X The precedingexpression is analogous to afuzzycomposition performedon fuzzyvectors, ◦ orb = a R, or in vector form, ∼ ∼ ∼ b = max(min(a ,r )) (12.14) i ij ∼j i whereb is the jth element of the fuzzy image B. ∼ ∼j Example12.2. Supposewehaveafuzzymapping,f,givenbythefollowingfuzzyrelation,R: ∼ ∼ 1.41.51.61.71.8 (m)   10.80.20.10 40 (kg) 0.810.80.20.1 50   R = 0.20.810.80.2 60   ∼   0.10.20.810.8 70 00.10.20.81 80 www.MatlabSite.com450 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE which represents a fuzzy mapping between the length and mass of test articles scheduled for flight in a space experiment. The mapping is fuzzy because of the complicated relationship between mass and the cost to send the mass into space, the constraints on length of the test articlesfittedintothecargosectionofthespacecraft,andthescientificvalueoftheexperiment. Suppose a particular experiment is being planned for flight, but specific mass requirements have not been determined. For planning purposes the mass (in kilograms) is presumed to be a fuzzy quantity described by the followingmembership function:   0.8 1 0.6 0.2 0 A = + + + + kg ∼ 40 50 60 70 80 or as a fuzzy vectora=0.8,1,0.6,0.2,0 kg. ∼ The fuzzy image B can be found using the extension principle (or, equivalently, ∼ ◦ composition for this fuzzy mapping), b = a R (recall that a set is also a one-dimensional ∼ ∼ ∼ relation). This composition results in a fuzzy output vector describing the fuzziness in the length of the experimental object (in meters), to be used for planning purposes, or b=0.8,1,0.8,0.6,0.2 m. ∼ PracticalConsiderations Heretofore we have discussed features of fuzzy sets on certain universes of discourse. Suppose there is a mapping between elements, u, of one universe, U, onto elements, v,of another universe, V, through a function f. Let this mapping be described by f : u → v. Define A to be a fuzzy set on universe U; that is, A ⊂ U. This relation is described by the ∼ ∼ membership function   µ µ µ 1 2 n A = + +···+ (12.15) ∼ u u u 1 2 n Then the extension principle, as manifested in Eq. (12.3), asserts that, for a function f that performs a one-to-one mapping (i.e., maps one element in universe U to one element in universe V), an obvious consequence of Eq. (12.3) is µ µ µ 1 2 n f(A) = f + +···+ ∼ u u u 1 2 n (12.16)   µ µ µ 1 2 n = + +···+ f(u ) f(u ) f(u ) 1 2 n The mapping in Eq. (12.16) is said to be one-to-one. Example12.3. Let a fuzzy set A be defined on the universe U=1,2,3.Wewishtomap ∼ elements of this fuzzy set to another universe, V, under the function v = f(u) = 2u −1 We see that the elements of V are V=1,3,5. Suppose the fuzzy set A is given by ∼   0.6 1 0.8 A = + + ∼ 1 2 3 www.MatlabSite.comEXTENSION PRINCIPLE 451 Then the fuzzy membership function for v = f(u) = 2u −1wouldbe   0.6 1 0.8 f(A) = + + ∼ 1 3 5 For cases where this functional mapping f maps products of elements from two universes, say U and U , to another universe V, and we define A as a fuzzy set on the 1 2 ∼ Cartesian space U ×U ,then 1 2   minµ (i), µ (j) 1 2 f(A) = i ∈ U ,j ∈ U (12.17) 1 2 ∼ f(i,j) where µ (i) and µ (j) are the separable membership projections of µ(i,j) from the 1 2 Cartesian space U ×U when µ(i,j) cannot be determined. This projection involves the 1 2 invocation of a condition known as noninteraction (see Chapter 2) between the separate universes.Itisanalogoustotheassumptionofindependenceemployedinprobabilitytheory, which reduces a joint probability density function to the product of its separate marginal density functions. In the fuzzy noninteraction case we are doing a kind of intersection; hence, we use the minimum operator (some logics use operators other than the minimum operator) as opposed to the product operator used in probability theory. Example12.4. Suppose we have the integers 1 to 10 as the elements of two identical but different universes; let U = U =1,2,3,...,10 1 2 Then define two fuzzy numbers A and B on universe U and U , respectively: 1 2 ∼ ∼   0.6 1 0.8 Define A = 2 = ‘‘approximately 2’’ = + + ∼ ∼ 1 2 3   0.8 1 0.7 Define B = 6 = ‘‘approximately 6’’ = + + ∼ ∼ 5 6 7 The product of (‘‘approximately 2’’) ×(‘‘approximately 6’’) should map to a fuzzy number ‘‘approximately 12,’’ which is a fuzzy set defined on a universe, say V, of integers, V = 5,6,...,18,21, as determined by the extension principle, Eq. (12.7), or 0.6 1 0.8 0.8 1 0.7 2 ×6 = + + × + + ∼ ∼ 1 2 3 5 6 7   min(0.6,0.8) min(0.6,1) min(0.8,1) min(0.8,0.7) = + +···+ + 5 6 18 21   0.6 0.6 0.6 0.8 1 0.7 0.8 0.8 0.7 = + + + + + + + + 5 6 7 10 12 14 15 18 21 In this example each of the elements in the universe, V, is determined by a unique mapping of the input variables. For example, 1 ×5 = 5, 2 ×6 = 12, etc. Hence, the maximum operation expressedinEq. (12.7)isnotnecessary.Itshouldalsobenotedthattheresultofthisarithmetic product is not convex, and hence does not appear to be a fuzzy number (i.e., normal and convex).However,thenonconvexityarisesfromnumerical aberrationsfromthediscretization of the two fuzzy numbers, 2 and 6, and not from any inherent problems in the extension ∼ ∼ principle. This issue is discussed at length later in this chapter in Example 12.14. www.MatlabSite.com452 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE The complexity of the extension principle increases when we consider if more than one of the combinations of the input variables, U and U , aremapped to the same variable 1 2 in the output space, V, i.e., if the mapping is not one-to-one. In this case we take the maximum membership grades of the combinations mapping to the same output variable, or, for the following mapping, we get µ (u ,u ) = max minµ (u ), µ (u ) (12.18) A 1 2 1 1 2 2 ∼ v=f(u ,u ) 1 2 Example12.5. We havetwofuzzy setsAand B,each defined onitsownuniverseasfollows: ∼ ∼     0.2 1 0.7 0.5 1 A = + + and B = + ∼ ∼ 1 2 4 1 2 We wish to determine the membership values for the algebraic product mapping f(A,B) = A ×B (arithmetic product) ∼ ∼ ∼ ∼  min(0.2,0.5) maxmin(0.2,1),min(0.5,1) = + 1 2  maxmin(0.7,0.5),min(1,1) min(0.7,1) + + 4 8   0.2 0.5 1 0.7 = + + + 1 2 4 8 In this case, the mapping involves two ways to produce a 2 (1 ×2and2 ×1) and two ways to produce a 4 (4 ×1and2 ×2); hence the maximum operation expressed in Eq. (12.7) is necessary. The extension principle can also be useful in propagating fuzziness through general- ized relations that are discrete mappings of ordered pairs of elements from input universes to ordered pairs of elements in an output universe. Example12.6. We want to map ordered pairs from input universes X =a,b and X = 1 2 1,2,3 to an output universe, Y=x,y,z. The mapping is given by the crisp relation, R, 12 3   ax z x R = bx y z We note that this relation represents a mapping, and it does not contain membership values. We define a fuzzy set A on universe X and a fuzzy set B on universe X as 1 2 ∼ ∼     0.6 1 0.2 0.8 0.4 A = + and B = + + ∼ ∼ a b 1 2 3 We wish to determine the membership function of the output, C = f(A,B), whose relational ∼ ∼ ∼ mapping, f, is described by R. This is accomplished with the extension principle, Eq. (12.7), www.MatlabSite.comEXTENSION PRINCIPLE 453 as follows: µ (x) = maxmin(0.2,0.6),min(0.2,1),min(0.4,0.6) = 0.4 C ∼ µ (y) = maxmin(0.8,1) = 0.8 C ∼ µ (z) = maxmin(0.8,0.6),min(0.4,1) = 0.6 C ∼ Hence,   0.4 0.8 0.6 C = + + ∼ x y z The extension principle is also useful in mapping fuzzy inputs through continuous- valued functions. The process is the same as for a discrete-valued function, but the effort involved in the computations is more rigorous. Example12.7 Wong and Ross, 1985. Suppose we have a nonlinear system given by the harmonic function x = cos(ωt), where the frequency of excitation, ω, is a fuzzy variable ∼ ∼ ∼ describedbythemembershipfunctionshowninFig. 12.2a.Theoutputvariable, x,willbefuzzy ∼ because of the fuzziness provided in the mapping from the input variable, ω. This function ∼ µ (ω) ω 1 ω ω 0 ω 1 2 supp ω (a) µ (ωt) ω t 1 t = ∆ t 0 ω ∆ t ω ∆ t ωt 1 2 supp ω t (b) µ (x) x x ω ∆ t 2 1 ω ∆ t 1 x –1 01 0 cos(ω ∆ t) cos(ω ∆ t) x = cos(ωt) 2 1 supp x (c)(d) FIGURE12.2 Extension principle applied to x = cos(ωt),at t = t. ∼ ∼ www.MatlabSite.com454 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE represents a one-to-one mapping in two stages, ω → ωt → x. The membership function of x ∼ ∼ ∼ ∼ willbe determined throughthe use ofthe extensionprinciple,whichfor thisexample willtake on the following form:  µ (x) = µ (ω) x ω ∼ ∼ x=cos(ωt) To show the development of this expression, we will take several time points, such as t = 0,1,....For t = 0, all values of ω map into a single point in the ωt domain, i.e., ωt = 0, ∼ ∼ ∼ and into a single point in the x universe, i.e., x = 1. Hence, the membership of x is simply a ∼ singleton at x = 1, i.e., 1, if x = 1 µ (x) = x ∼ 0, otherwise Foranonzerobutsmall t,say t = t,thesupportof ω,denotedinFig. 12.2aas supp ω, ∼ ∼ is mapped into a small but finite support of x, denoted in Fig. 12.2c as supp x (the support of ∼ ∼ + a fuzzy set was defined in Chapter 4 as the interval corresponding to a λ-cut of λ = 0 ). The membership value for each x in this interval is determined directly from the membership of ω ∼ inaone-to-onemapping.AscanbeseeninFig. 12.2,as t increases,thesupportofxincreases, ∼ andthefuzzinessintheresponsespreadswithtime.Eventually,therewillbeavalueof t when µ (ω) ω 1 0 ω ω ω 1 2 supp ω (a) µ (ωt) ω t 1 0 ω t ω t ωt 1 2 supp ωt (b) t ω 1 µ (x) x x 1 –1 0 1 x ω t 2 –1 cos(ω t) cos(ω t) 0 x = cos(ωt) 2 1 supp x (c) (d) FIGURE12.3 Extension principle applied to x = cos(ωt) when t causes overlap in support of x. ∼ ∼ ∼ www.MatlabSite.comFUZZY ARITHMETIC 455 µ (ω) ω 1 0 ω ω ω 1 2 (a) µ (ωt) ω t 1 0 ω t ω t ωt 1 2 (b) µ (x) x ω t 1 1 ω t 2 –1 0 1 x –1 0 1 x = cos(ωt) cos(ω t) cos(ω t) 1 2 (c) (d) FIGURE12.4 Extension principle applied to x = cos(ωt) when t causes complete fuzziness. ∼ ∼ the support of x folds partly onto itself, i.e., we have multi-ω-to-single-x mapping. In this ∼ event,themaximum of allcandidatemembership values of ω isusedas themembership value ∼ of x according to the extension principle, Eq. (12.3), as shown in Fig. 12.3(c). When t is of such magnitude that the support of x occupies the interval −1,1 completely, the largest support possible, the membership µ (x) will be unity for all x within x ∼ this interval. This is the state of complete fuzziness, as illustrated in Fig. 12.4. In the equation x = cos(ωt), the output can have any value in the interval −1,1 with equal and complete ∼ ∼ membership. Once this state is reached the system remains there for all future time. FUZZYARITHMETIC Chapter 4 defines a fuzzy number as being described by a normal, convex membership functionontherealline;fuzzynumbers usuallyalsohavesymmetricmembershipfunctions. In this chapter we wish to use the extension principle to perform algebraic operations on fuzzy numbers (as illustrated in previous examples in this chapter). We define a normal, convex fuzzy set on the real line to be a fuzzy number, and denote it I. ∼ www.MatlabSite.com456 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE Let I and J be two fuzzy numbers, with I defined on the real line in universe X and ∼ ∼ ∼ J defined on the real line in universe Y, and let the symbol ∗ denote a general arithmetic ∼ operation, i.e., ∗≡+, −, ×, ÷. An arithmetic operation (mapping) between these two number, denoted I ∗J, will be defined on universe Z, and can be accomplished using the ∼ ∼ extension principle, by  µ (z) = (µ (x) ∧ µ (y)) (12.19) I∗J I J ∼ ∼ ∼ ∼ x∗y=z Equation (12.19) results in another fuzzy set, the fuzzy number resulting from the arithmetic operation on fuzzy numbers I and J. ∼ ∼ Example12.8. We want to perform a simple addition (∗≡+) of two fuzzy numbers. Define a fuzzy one by the normal, convex membership function defined on the integers,   0.2 1 0.2 1 = + + ∼ 0 1 2 Now,wewanttoadd‘‘fuzzyone’’plus‘‘fuzzyone,’’usingtheextensionprinciple,Eq. (12.19), to get 0.2 1 0.2 0.2 1 0.2 1 +1 = 2 = + + + + + ∼ ∼ ∼ 0 1 2 0 1 2  min(0.2,0.2) maxmin(0.2,1),min(1,0.2) = + 0 1 maxmin(0.2,0.2),min(1,1),min(0.2,0.2) + 2  maxmin(1,0.2),min(0.2,1) min(0.2,0.2) + + 3 4   0.2 0.2 1 0.2 0.2 = + + + + 0 1 2 3 4 Note that there are two ways to get the resulting membership value for a 1 (0 +1and1 +0), three ways to get a 2 (0 +2,1 +1,2 +0),andtwowaystogeta3 (1 +2and2 +1).These are accounted for in the implementation of the extension principle. The support for a fuzzy number, say I (see Chapter 4), is given by ∼ supp I=x µ (x) 0=I(12.20) I ∼ ∼ whichisanintervalontherealline,denotedsymbolicallyasI.SinceapplyingEq. (12.19)to arithmeticoperationsonfuzzynumbersresultsinaquantitythatisalsoafuzzynumber,we can find the support of the fuzzy number resulting from the arithmetic operation, I ∗J, i.e., ∼ ∼ supp(z) = I ∗J (12.21) I∗J ∼ ∼ whichisseentobethearithmeticoperationonthetwoindividualsupports(crispintervals), I and J, for fuzzy numbers I and J, respectively. ∼ ∼ www.MatlabSite.comINTERVAL ANALYSIS IN ARITHMETIC 457 + Chapter 4revealedthatthesupportofafuzzysetisequaltoits λ-cutvalueat λ = 0 . In general, we can perform λ-cut operations on fuzzy numbers for any value of λ.A result we saw in Chapter 4 for set operations (Eq. (4.1)) is also valid for general arithmetic operations:   I ∗J = I ∗J (12.22) λ λ ∼ ∼ λ Equation 12.22 shows that the λ-cut on a general arithmetic operation (∗≡+, −, ×, ÷) on two fuzzy numbers is equivalent to the arithmetic operation on the respective λ-cuts of thetwofuzzynumbers.Both (I ∗J) andI ∗J areintervalquantities;andmanipulationsof λ λ λ ∼ ∼ these quantities can make use of classical interval analysis, the subject of the next section. INTERVALANALYSISINARITHMETIC As alluded to in Chapter 2, a fuzzy set can be thought of as a crisp set with ambiguous boundaries. In this sense, as Chapter 4 illustrated, a convex membership function defining a fuzzy set can be described by the intervals associated with different levels of λ-cuts. Let I and I be two interval numbers defined by ordered pairs of real numbers with lower and 1 2 upper bounds: I = a,b, where a ≤ b 1 I = c,d, where c ≤ d 2 When a = b and c = d, these interval numbers degenerate to a scalar real number. We again define a general arithmetic property with the symbol ∗,where ∗≡+, −, ×, ÷. Symbolically, the operation I ∗I = a,b ∗c,d (12.23) 1 2 represents another interval. This interval calculation depends on the magnitudes and signs of the elements a, b, c,and d. Table 12.1 shows the various combinations of set-theoretic intersection (∩) and set-theoretic union (∪) for the six possible combinations of these elements (ab and cd still hold). Based on the information in Table 12.1, the four arithmetic interval operations associated with Eq. (12.23) are given as follows: a,b +c,d = a + c,b + d (12.24) a,b −c,d = a − d,b − c (12.25) a,b ·c,d = min(ac,ad,bc,bd),max(ac,ad,bc,bd) (12.26)   1 1 a,b ÷c,d = a,b · , provided that 0 ∈ / c,d (12.27) d c  αa,αbforα 0 αa,b = (12.28) αb,αaforα 0 where ac, ad, bc,and bd are arithmetic products and 1/d and 1/c are quotients. The caveat applied to Eq. (12.27) is that the equivalence stated is not valid for the case when c ≤ 0 and d ≥ 0 (obviously the constraintcd still holds), i.e., zero cannot be contained within the interval c,d. Interval arithmetic follows properties of associativity www.MatlabSite.com458 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE TABLE12.1 Set operations on intervals Cases Intersection(∩)Union(∪) ad ∅ c,d ∪a,b cb ∅ a,b ∪c,d ac,bd a,bc,d ca,d b c,da,b acbd c,ba,d ca d b a,dc,b and commutativity for both summations and products, but it does not follow the property of distributivity. Rather, intervals do follow a special subclass of distributivity known as subdistributivity, i.e., for three intervals, I, J, and K, I · (J +K) ⊂ I ·J +I ·K (12.29) The failure of distributivity to hold for intervals is due to the treatment of two occurrences of identical interval numbers (i.e., I) as two independent interval numbers Dong and Shah, 1987. Example12.9. −3 ·1,2 = −6,−3 0,1 −0,1 = −1,1 1,3 ·2,4 = min(2,4,6,12),max(2,4,6,12) = 2,12 1 1 1,2 ÷1,2 = 1,2 · ,1 = ,2 2 2 Considerthefollowingexampleofsubdistributivity.ForI = 1,2,J = 2,3,K = 1,4,then I · (J −K) = 1,2 · (2,3 −1,4) = 1,2 ·−2,2 = −4,4 I ·J −I ·K = 1,2 ·2,3 −1,2 ·1,4 = 2,6 −1,8 = −6,5 Now, −4,4 = −6,5, but −4,4 ⊂ −6,5. Interval arithmetic can be thought of in the following way. When we add or multiply two crisp numbers, the result is a crisp singleton. When we add or multiply two intervals we are essentially performing these operations on the infinite number of combinations of pairs of crisp singletons from each of the two intervals; hence, in this sense, an interval is expectedastheresult.Inthesimplestcase,whenwemultiply twointervalscontainingonly positiverealnumbers,itiseasyconceptuallytoseethattheintervalcomprisingthesolution is found by taking the product of the two lowest values from each of the intervals to form the solution’s lower bound, and by taking the product of the two highest values from each of the intervals to form the solution’s upper bound. Even though we can see conceptually that an infinite number of combinations of products between these two intervals exist, we need only the endpoints of the intervals to find the endpoints of the solution. www.MatlabSite.comAPPROXIMATE METHODSOF EXTENSION 459 APPROXIMATEMETHODSOFEXTENSION A serious disadvantage of the discretized form of the extension principle in propagating fuzziness for continuous-valued mappings is the irregular and erroneous membership functions determined for the output variable if the membership functions of the input variables are discretized for numerical convenience (this problem is demonstrated in Example 12.14). The reason for this anomaly is that the solution to the extension principle, as expressed in Eq. (12.7), is really a nonlinear programming problem for continuous- valued functions. It is well-known that, in any optimization process, discretization of any variables canlead to an erroneous optimum solution becauseportions of the solution space are omitted in the calculations. For example, try to plot a 10th-order curve with a series of equally spaced points; some local minimum and maximum points on the curve are going to be missed if the discretization is not small enough. Again, these problems do not arise because of any inherent problems in the extension principle itself; they arise whencontinuous-valuedfunctionsarediscretized,thenallowedtopropagatefromtheinput domain to the output domain using the extension principle. Othermethodshavebeenproposedtoeasethecomputationalburdeninimplementing the extension principle for continuous-valued functions and mappings. Among the alter- native methods proposed in the literature to avoid this disadvantage for continuous fuzzy variables are three approaches that are summarized here along with illustrative numerical examples. All of these approximate methods make use of intervals, at various λ-cut levels, in defining membership functions. VertexMethod A procedure known as the vertex method Dong and Shah, 1987 greatly simplifies manipulations of the extension principle for continuous-valued fuzzy variables, such as fuzzy numbers defined on the real line. The method is based on a combination of the λ-cut concept and standard interval analysis. The vertex method can prevent abnormality in the output membership function due to application of the discretization technique on the fuzzy variables’ domain, and it can prevent the widening of the resulting function value set due to multiple occurrences of variables in the functional expression by conventional interval analysis methods. The algorithm is very easy to implement and can be computationally efficient. The algorithm works as follows. Any continuous membership function can be + represented by a continuous sweep of λ-cut intervals from λ = 0 to λ = 1. Figure 12.5 shows a typical membership function with an interval associatedwith a specific value of λ. Supposewehaveasingle-inputmappinggivenby y = f(x)thatistobeextendedforfuzzy sets, or B = f(A), and we want to decompose A into a series of λ-cut intervals, say I . λ ∼ ∼ ∼ When the function f(x) is continuous and monotonic on I = a,b, the interval λ representing B at a particular value of λ,say B , can be obtained by λ ∼ B = f(I ) = min(f (a), f (b)),max(f (a), f (b)) (12.30) λ λ Equation (12.30) has reduced the interval analysis problem for a functional mapping to a simple procedure dealing only with the endpoints of the interval. When the mapping www.MatlabSite.com460 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE µ 1 A λ I λ 0 ab x FIGURE12.5 Interval corresponding to a λ-cut level on fuzzy set A. ∼ x 3 (a , a , b)(a , b , b ) 1 2 3 1 2 3 (b , a , b ) 1 2 3 (b , b , b ) 1 2 3 (a , a , a ) 1 2 3 x (a , b , a ) 1 2 3 2 (b , a , a ) (b , b , a ) 1 2 3 1 2 3 x 1 FIGURE12.6 Three-dimensional Cartesian region involvingintervals for three input variables, x , x ,and x . 1 2 3 is given by n inputs, i.e., y = f(x ,x ,x ,...,x ), then the input space can be represented 1 2 3 n byan n-dimensional Cartesianregion; a3D Cartesianregionisshown in Fig. 12.6.Eachof the input variables can be described by an interval, say I ,ataspecific λ-cut, where iλ I = a ,b i = 1,2,...,n (12.31) iλ i i As seenin Fig. 12.6, the endpoint pairs of eachinterval given in Eq. (12.31) intersect in the 3D space and form the vertices (corners) of the Cartesian space. The coordinates of these vertices are the values used in the vertex method when determining the output n interval for each λ-cut. The number of vertices, N, is a quantity equal to N = 2 ,where n is the number of fuzzy input variables. When the mapping y = f(x ,x ,x ,...,x ) is 1 2 3 n continuous in the n-dimensional Cartesian region and when also there is no extreme point in this region (or along the boundaries), the value of the interval function for a particular www.MatlabSite.comAPPROXIMATE METHODSOF EXTENSION 461 λ-cut can be obtained by   B = f(I ,I ,I ,...,I ) = min(f (c )),max(f (c )) j = 1,2,...,N (12.32) λ 1λ 2λ 3λ nλ j j j j where c isthecoordinateofthe jthvertexrepresentingthe n-dimensionalCartesianregion. j Thevertexmethodisaccurateonlywhenthe conditionsofcontinuity andnoextreme point are satisfied. When extreme points of the function y = f(x ,x ,x ,...,x ) exist in 1 2 3 n the n-dimensional Cartesian region of the input parameters, the vertex method will miss certainpartsoftheintervalthatshouldbeincludedintheoutputintervalvalue,B .Extreme λ points can be missed, for example, in certain mappings that are not one-to-one. If the extreme points can be identified, they are simply treated as additional vertices, E ,inthe k Cartesian space and Eq. (12.32) becomes, because the continuity property still holds,   B = min(f (c ), f (E )),max(f (c ), f (E )) (12.33) λ j k j k j,k j,k where j = 1,2,...,N and k = 1,2,...,m for m extreme points in the region. Example12.10. We wish to determine the fuzziness in the output of a simple nonlinear mapping given by the expression y = f(x) = x(2 − x), seen in Fig. 12.7a, where the fuzzy input variable, x, has the membership function shown in Fig. 12.7b. We shall solve this problem using the fuzzy vertex method at three λ-cut levels, for + λ = 0 ,0.5,1. As seen in Fig. 12.7b, the intervals corresponding to these λ-cuts are I + = 0 0.5,2,I = 0.75,1.5,I = 1,1 (a single point). Since the problem is one-dimensional, .5 1 1 thevertices, c ,aredescribedbyasinglecoordinate;thereare N = 2 = 2vertices (j = 1,2). j In addition, an extreme point does exist within the region of the membership function and is determined using a derivative of the function, df (x)/dx = 2 −2x = 0,x = E =1(E , 0 1 k where k = 1).Thisextremepointiswithineachofthethree λ-cutintervals,sowillbeinvolved in all the following calculations for B : λ I + = 0.5,2 0 c = 0.5,c = 2,E = 1 1 2 1 y µ µ(x) 1 1 0.75 A 0.5 0 0.5 1 2 x 0 0.5 1 2 x 0.75 1.5 (a)(b) FIGURE12.7 Nonlinear function and fuzzy input membership. www.MatlabSite.com462 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE f(c ) = 0.5(2 −0.5) = 0.75,f(c ) = 2(2 −2) = 0, 1 2 f(E ) = 1(2 −1) = 1 1 + B = min(0.75,0,1),max(0.75,0,1) = 0,1 0 I = 0.75,1.5 0.5 c = 0.75,c = 1.5,E = 1 1 2 1 f(c ) = 0.75(2 −0.75) = 0.9375,f(c ) = 1.5(2 −1.5) = 0.75, 1 2 f(E ) = 1(2 −1) = 1 1 B = min(0.9375,0.75,1),max(0.9375,0.75,1) = 0.75,1 0.5 I = 1,1 1 c = 1,c = 1,E = 1 1 2 1 f(c ) = f(c ) = f(E ) = 1(2 −1) = 1 1 2 1 B = min(1,1,1),max(1,1,1) = 1,1 = 1 1 Figure 12.8 provides a plot of the intervals B +,B ,andB to form the fuzzy output, y. 0 0.5 1 DSWAlgorithm TheDSWalgorithmDong,Shah,andWong,1985alsomakesuseofthe λ-cutrepresenta- tionoffuzzysets,but,unlikethevertexmethod,itusesthefull λ-cutintervalsinastandard interval analysis. The DSW algorithm consists of the following steps: 1. Select a λ value where 0 ≤ λ ≤ 1. 2. Find the interval(s) in the input membership function(s) that correspond to this λ. 3. Using standard binary interval operations, compute the interval for the output member- ship function for the selected λ-cut level. 4. Repeat steps 1–3 for different values of λ to complete a λ-cut representation of the solution. µ(y) 1 B 0.5 0 0.5 0.75 1 y FIGURE12.8 Fuzzy membership function for the output to y = x(2 −x). www.MatlabSite.comAPPROXIMATE METHODSOF EXTENSION 463 Example12.11. Let us consider a nonlinear, 1D expression similar to the previous example, 2 or y = x(2 + x) = 2x + x , where we again use the fuzzy inputvariable shown in Fig. 12.7b. ThenewfunctionisshowninFig. 12.9a,alongwiththefuzzyinputinFig. 12.9b.Again,ifwe + decompose the membership function for the input into three λ-cut intervals, for λ = 0 ,0.5, + and 1, we get the intervalsI = 0.5,2, I = 0.75,1.5, and I = 1,1 (a singlepoint).In 0 0.5 1 terms of binary interval operations, the functional mapping on the intervals would take place as follows for each λ-cut level: + I = 0.5,2 0 2 2 B + = 20.5,2 +0.5 ,2 = 1,4 +0.25,4 = 1.25,8 0 I = 0.75,1.5 0.5 2 2 B = 20.75,1.5 +0.75 ,1.5 = 1.5,3 +0.5625,2.25 = 2.0625,5.25 0.5 I = 1,1 1 2 2 B = 21,1 +1 ,1 = 2,2 +1,1 = 3,3 = 3 1 Figure 12.10 provides a plot of the intervals B +,B ,andB to form the fuzzy output, y. 0 0.5 1 y µ 8 1 A 3 01 2 x 0 0.5 1 2 x (a)(b) FIGURE12.9 Nonlinear function and fuzzy input membership. µ 1 B 0.5 01 2 4 6 8 y FIGURE12.10 Fuzzy membership function for the output to y = x(2 +x). www.MatlabSite.com464 FUZZY ARITHMETIC AND THE EXTENSION PRINCIPLE The previous example worked with a fuzzy input that was defined on the positive sideoftherealline;henceDSWoperationswereconductedonpositivequantities.Suppose we want to conduct the same DSW operations, but on a fuzzy input that is defined on both the positive and negative side of the real line. The user of the DSW algorithm must be careful in this case. If the lower bound of an interval is negative and the upper bound is positive (i.e., if the interval contains zero) and if the function involves a square or an even-power operation, then the lower bound of the result should be zero. This feature of an interval analysis, like the DSW method, is demonstrated in the following example. Example12.12. Let us consider the nonlinear, 1D expression from Example 12.11, i.e., 2 y = x(2 + x) = 2x + x ,whichisshowninFig. 12.9aandisrepeatedinFig. 12.11a.Suppose we change the domain of the input variable, x, to include negative numbers, as shown in Fig. 12.11b. Again, if we decompose the membership function for the input into three λ-cut + intervals,for λ = 0 , 0.5, and 1,we get the intervals I + = −0.5,1, I = −0.25,0.5, and 0 0.5 I = 0,0 (a single point). In terms of binary interval operations, the functional mapping on 1 the intervals would take place as follows for each λ-cut level: + I = −0.5,1 0 2 B + = 2−0.5,1 +0,1 = −1,2 +0,1 = −1,3 0 2 (Note:Theboldfacezeroistakenastheminimum,since (−0.5) 0;becausezeroiscontained in the interval −0.5,1 the minimum of squares of any number in the interval will be zero.) I = −0.25,0.5 0.5 2 B = 2−0.25,0.5 +0,0.5 = −0.5,1 +0,0.25 = −0.5,1.25 0.5 I = 0,0 1 B = 20,0 +0,0 = 0,0 1 + Figure 12.12 is a plot of the intervals B ,B ,andB that form the fuzzy output, y. 0 0.5 1 y µ(x) 8 1.0 A 3 01 2 x −0.5 0 1.0 x (a)(b) FIGURE12.11 Nonlinear function and fuzzy input membership. www.MatlabSite.com

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